Method and apparatus for trimming B-spline descriptions of patches in a high performance three dimensional graphics system

ABSTRACT

A graphics accelerator responds to commands from a computer in a graphic system by storing the definitions of nonuniform rational B-spline patches and their associated trimming curves. The graphics accelerator then produces device coordinates for trimmed polygons computed for each patch and sends these polygons to a display. The B-spline definitions of the trimming curves in the uv parameter space of each patch are converted to approximating short straight line segments. Untrimmed polygon vertices, the end points of the straight line segments and the intersections of the straight line segments with subspan boundaries corresponding to polygon edges are kept in a data structure of linked lists of vertex tables. The data structure is traversed to determine new polygon vertices for trimmed polygons. The trimming mechanism is compatible with recursive subdivision of patches to overcome practical limitations on the number of trimming curves that may be associated with each patch. The length of the straight line segments of the trimming curves is adjusted to compensate for less than ideal parameterization of the trimming curve functions. Associated with each trimming curve within a patch is information about the position of that trimming curve in the span. As each polygon for that patch is generated, those trimming curves that are clearly outside the clip limits for that polygon are excluded from consideration.

BACKGROUND AND SUMMARY OF THE INVENTION

A typical high performance three dimensional graphics system willdescribe a surface to be rendered as surface patches defined byfunctions for each patch. Such functions might be, for example,nonuniform rational B-splines. The use of B-splines imposes certainlimitations upon the edges of surface patches. Associated with B-splinefunctions is a normally rectangular uv parameter space. Parametric patchgeneration functions of u and v compute the values of the coordinates inXYZ space. The rectangular uv space limits the exactness with which theB-spline patch generation functions can represent surface patches havingedges of certain shapes. For example, it is difficult to produce a goodB-spline description for a patch that is a rectangular region with acircular portion removed from its interior. Either of the rectangularregion or the circular portion by themselves would be practical, buttheir combination is too complex a primitive for a single unifiedB-spline description at the patch level. Subdividing the patch thwartsthe motive for having patches in the first place. Trimming is a way toaugment the B-spline description of the rectangular region with anotherone for the circular portion, and producing a hybrid surface patch inwhich one B-spline description "trims away" the surface described byanother.

In the prior art, trimming has been performed by the software of thegraphics system prior to the sending of device coordinates to thedisplay hardware. Such trimming is necessarily a very complex task, andis generally too slow for use with moving images or interactive systems.It would be desirable to retain the use of B-splines and achieve theadvantages offered by that technique of surface description, but at thesame time allow high speed trimming.

According to a preferred method of the invention, trimming is performedon B-spline surface patch descriptions in a hardware graphicsaccelerator. It receives B-spline descriptions of the patch generationfunctions for the untrimmed patches and B-spline descriptions oftrimming curves in the uv parameter space of the patch generationfunctions. The B-spline descriptions of the trimming curves arethemselves functions of a parameter t. The graphics accelerator computesa sufficiently dense point by point representation of each trimmingcurve in uv space, in addition to point by point representations of theindividual subspans in uv space whose associated polygons in XYZ spaceapproximate the patch. The graphics accelerator determines wherestraight line approximating segments of the trimming curves crosssubspan boundaries and changes the vertices of the subspans to trim awayportions of the associated polygon. It does this by building a datastructure of linked lists of vertex tables that represent the untrimmedpolygon and any trimming curves that cut it. An appropriate traversal ofthe lists in the data structure produces a list of trimmed polygonvertices in device coordinates That list may then be further processedto the pixel level by other hardware in the graphics accelerator.

Considerable attention is paid to avoiding the evils of roundoff error.A mechanism for describing where in a span a trimming curve is locatedallows the trimming operation to exclude from consideration thosetrimming curves that cannot possibly affect the polygon being processed.Another mechanism compensates for the effects of a non-idealparameterization of the trimming curve functions, to prevent theproduction of an unnecessarily high number of polygon vertices. Thetrimming method is also compatible with recursive subdivision of patchesto handle patches with a high number of trimming curves.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an illustration of a surface in XYZ space approximated bypatches formed by mapping spans of a two dimensional uv parameter spaceinto XYZ space with parametric patch generation functions, such asnonuniform rational B-splines

FIG. 2 shows how a parametric function of a one dimensional t space candefine a trimming curve in uv space that in turn can be mapped into XYZspace to identify a region that is absent from the surface.

FIG. 3 illustrates a parametric trimming curve function segmented overseveral spans of the one dimensional t space and defining a trimmingcurve in the two dimensional uv space.

FIG. 4 illustrates how a segmented trimming curve function can crossseveral spans of uv space to define a mapped trimming curve in acorresponding number of patches in XYZ space.

FIG. 5 illustrates the segmented trimming curve of FIG. 4 in greaterdetail.

FIG. 6 shows how the spans in uv space are divided into subspans toproduce surface approximating polygons for the surface in XYZ space, andalso indicates that the trimming curve in uv space crosses the subspanswhile the mapped trimming curve trims the polygons on the surface.

FIG. 7 is an enlarged view of a portion of FIG. 6, illustrating thepresence of extra vertices along patch boundaries and along mappedtrimming curves for better patch-to-patch transitions, and the presenceof points along the mapped trimming curve defining points ofintersection with the polygon edges as well as approximating pointswithin the interior regions of polygons crossed by the mapped trimmingcurve.

FIG. 8 is an expansion of a portion of FIG. 7 showing how the shapes ofpolygons are changed by being crossed by a mapped trimming curve.

FIGS. 9A-C comprise a simplified section of pseudocode describing theprocessing steps used to produce polygons crossed by mapped trimmingcurves and whose shapes have been adjusted accordingly, and include anillustration of the structure of a polygon vertex table whose instancesare linked together into lists describing polygons.

FIG. 10 depicts the order within a span in which subspans are taken togenerate the approximating polygons for the surface in XYZ space.

FIG. 11 illustrates several types of trimming curves defined upon aspan, and indicates the presence of a guard region around the span toassist in avoiding arithmetic rounding errors.

FIGS. 12A-B depict how a clipping mechanism operates upon approximatingstraight line segments of trimming curves that cross subspans to producepoints of intersection between the straight line segments of thetrimming curve and the boundaries of the subspan.

FIGS. 13A-B describe certain special cases when trimming curves exhibitpoints in uv space that are very close to the boundaries of the subspanof interest.

FIG. 14 illustrates a useful example of a subsapn in uv space crossed bycertain particular trimming curves, and serves as the basis of anexplanation of how the linked boundary list of vertex tables and itslinked trimming curve sublists are traversed to produce polygons thatare present ON the surface.

FIG. 15 is a simplified schematic representation of a linked boundarylist and trimming curve sublists for the vertex tables corresponding tothe example of FIG. 14.

FIGS. 16A-C comprise a simplified flowchart for processing the linkedlist of vertex tables in accordance with step 12 of FIG. 9C.

FIG. 17 is a simplified section of pseudocode pertaining to the samesubject matter as FIGS. 16A-C, but with more detail concerning thestructure of the vertex tables shown in FIG. 9B and FIG. 15.

FIG. 18 is a diagram of a trimming curve in uv space illustrating arefinement in the selection of points in uv space representing thetrimming curve in the operations of clipping and trimming.

FIG. 19 is a simplified hardware block diagram of the componentinstruments used in an embodiment incorporating the preferred method ofthe invention.

FIGS. 20A-C comprise a simplified hardware block diagram of the graphicsaccelerator of FIG. 19.

DESCRIPTION OF A PREFERRED METHOD

We shall begin by considering one way that a three dimensional surfacemay be represented within a graphics system. Suppose, for example, thata saddle-shaped surface 1 such as the one depicted in FIG. 1 were to berepresented. This would be accomplished by dividing the entire surface 1into smaller portions (commonly called patches), of which the patch 2 isillustrative. To adequately render a surface as complicated as thesaddle shape 1 a great many patches would be required. For the sake ofsimplicity we begin by showing only one patch 2; it will be understoodthat the graphics system would include some means to divide the surfaceinto appropriate patches. Also, the various other curved lines in FIG. 1are not there to suggest other patches (although they might), but areinstead offered as an aid in appreciating the shape of the surface 1.Furthermore, it will be understood that, although the surface 1resembles a hyperbolic paraboloid, the method to be described isparticularly applicable to irregular ("free-form") surfaces, especiallythose rendered from parametric B-spline descriptions.

Associated with a patch is a portion 5 of a parameter space 3. Theportion is called a span. To describe a three dimensional surface in XYZspace a two dimensional paramenter space is employed. In the presentexample the two paramenters are u and v, and each is allowed to varybetween associated maximum and minimum values. As shown in the figure, aset of parametric patch generation equations 4 map values in theparameter space 3 into XYZ space to produce the (X,Y,Z) triples that lieon the patch 2.

A modern high performance graphics system will store a completedescription of the object, independent of any particular view that maybe desired. Often this description takes the form of a data base,particularly when subportions of the object are described separately andthen deployed by reference, perhaps a multiplicity of times Rather thanstore in the database an exhaustive collection of points on the surfaceit is common to instead employ a more compact type of description, suchas approximating functions that yield values to a desired degree ofresolution. When a particular view is wanted it is found by computingtranslations and or rotations for points first found by evaluation ofthe approximating functions in the database. B-splines are a techniquefor achieving such a compact representation through parametricfunctions.

Even an introductory explanation of B-splines is beyond what can beincluded here, and the reader is referred to the various reference workson the subject (e.g.: Fundamentals of Interactive Computer Graphics byJ. D. Foley and A. Van Dam, Addison-Wesley, 1984; and an articleentitled Rational B-Splines for Curve and Surface Representation, IEEEComputer Graphics and Applications, vol. 3 no. 6, Sept. 1983).Fortunately, it is not necessary here to understand in depth whatB-splines are and how they work. Various software packages exist toproduce B-spline descriptions of desired solid objects and surfaces. Thepoint to be made here is that, while the method of the invention lendsitself well to use with B-spline descriptions of solid objects andsurfaces, the use of B-splines is in no way required. However, since thepreferred method has been used in a graphics system that does employB-splines, it is most convenient to describe the method in that setting.That graphics system is the Hewlett-Packard 9000 Model 320SRX, whichincludes, in particular, an HP Model 98720A Graphics Accelerator.

Specifically, FIG. 1 illustrates that each point 6 on a surface ofinterest 1 can be represented by a corresponding point 7 in some portion5 of a parameter space 3. The user of the graphics system interacts withthat system in defined ways set out in an operating manual for thesystem. By using appropriate commands a desired surface may beconstructed. In general, the surface will be composed of a multiplicityof patches. For each patch upon a surface in XYZ space, the internalactivities of the graphics system produce three (or perhaps four) patchgeneration functions of u and v. The variables u and v belong to theparameter space 3. The x coordinate of the point 6 is found byevaluating a (polynomial) function F_(x) (u,v). Corresponding functionsF_(y) and F_(z) produce the y and z coordinates of the point 6. A moregeneral case (rational B-splines) for three coordinates uses fourfunctions:

    ______________________________________                                                   x = F.sub.x (u,v)/H(u,v)                                                      y = F.sub.y (u,v)/H(u,v)                                                      z = F.sub.z (u,v)/H(u,v)                                           ______________________________________                                    

In this case each of the functions F_(x), F_(y) and F_(z) has beenrationalized through division by the function H. The method to bedescribed is compatible with either rational or nonrational nonuniformB-spline representations.

Conceptually, one could compute all the pixel values for the surface byevaluating the B-spline patch generation functions for a sufficientlydense cartesian product of the variables u and v; i.e., by evaluating asufficiently dense collection of points in the parameter space 3. Inpractice, however, such an approach requires an inordinate amount ofexecution time, and is generally avoided in favor of one that requiresonly quicker and more easily performed computations. The polylgon methodto be described in connection with FIGS. 6-8 is such a favored approach.According to this approach the set of points evaluated in the parameterspace for a given patch are selected to be just dense enough to yieldpolygons vertices for polygons that, when interpolated by shading,produce an acceptably smooth surface. Since each polygon contains amultitude of pixels whose values are relatively easily found by linearinterpolation done in hardware, a considerable reduction in imagepreparation time is achieved.

It is unlikely that one set of functions 4 entirely describes a completesurface of interest. Instead, collections of B-spline patch generationfunctions are found that provide a good approximations of the desiredsurface over small regions. This piecewise approximation divides theparameter space into regions, called spans. The piecewise approximationis what actually divides the surface into the corresponding patches. Torender a patch, its corresponding B-spline functions are evaluated overthe range of the associated span. Acccording to the desired resolutionfor the displayed object (i.e., the polygon size), suitable step sizesare selected for increments to u and v. The parameter u is allowed torange in equal steps from u_(min) to u_(max), while the parameter vranges in equal steps (not necessarily the same as those for u) fromv_(min) to v_(max). This evaluation of the span produces "raw" polygonvertices that are generally subjected to a fair amount of furtherprocessing prior to being displayed.

Thus we see that the surface to be displayed is described as acollection of patches, each of which has an associated span. Associatedwith each span is a particular collection of B-spline patch generationfunctions.

Recall that it was said above that by using appropriate commands theuser can construct a desired surface. Suppose, for example, that a flatplate having a hole in it were the desired object. With a typical solidmodeling package a user would select from a menu a suitable primitiveshape, such as a rectangular solid, and specify its dimensions. He wouldthen specify the location of the center of the hole, and "subtract" fromthe plate a cylinder of the desired diameter. It is a practical matterto produce a B-spline description of a rectangular plate. It is also apractical matter to obtain a B-spline description of the cylinder to beremoved by "drilling" the hole. However, it is not a practical matter toconsider the plate with the hole already in it as a primitive shapehaving its own particular B-spline description.

The method of the invention affords a solution to the problem of complexsurfaces that do not have practical B-spline descriptions. Just as thehole in the plate has an edge in XYZ space, there are correspondingpoints in uv space which, when evaluated, will exactly describe orclosely approximate the edge of the material to be removed to producethe hole. These points lie along a curve in uv space, called a trimmingcurve. The trimming curve can be represented by one or more functions.These functions can be found by the solid modeling package. Thereafter,the rendering of a patch having a "subtracted" portion can beaccomplished by a qualified evaluation of the patch generation functionsfor the associated span. The qualification takes the form ofdetermining, during the evaluation of the patch generation functions,whether the (u,v) pair at hand lies on, inside or outside of the timmingcurve. The qualification produces a decision about whether or not todisplay the associated (X,Y,Z) triple. If that triple is to bedisplayed, things proceed as previously described. However, if theevaluation of that (u,v) pair would produce a polygon vertex lyingwithin the region to be subtracted, then the triple cannot be displayedand the existing polygon structure must be modified.

FIG. 2 illustrates the use of a trimming curve 8 to describe theboundary in XYZ space of a region 9 to be subtracted from patch 2. Thetrimming curve 8 is defined in a span 5 of the parameter space 3. In thepreferred method the trimming curve 8 can itself be defined byB-splines. A pair of parametric equations 9 of a parameter t can definea two dimensional trimming curve in uv space. After the fashion of thepatch generation functions, trimming curve functions 9 can be eitherrational or nonrational non uniform B-splines.

Just as a surface is composed of patches and their associatedcollections of different patch generation functions, a trimming curvemust generally be piecewise assembled from subportions called segments.Each segment has its own particular trimming curve functions 9.

This state of affairs is depicted in FIG. 3 for the trimming curve 8 ofFIG. 2. That trimming curve has been broken into a number of segments S₁through S₆. Note that each segment has its own functions that aredefined over corresponding spans of t space 10.

FIG. 4 describes a generalization of the application of trimming curves.What is shown is similar to the situation depicted in FIGS. 2 and 3, butwith the difference that the trimming curve 11 now traverses severalspans 12-15 in uv space. This situation arises because, as luck wouldhave it, the portion 9 to be subtracted from the surface 1 occupies thefour corresponding patches 16-19. The segment boundaries of the trimmingcurve 11 are shown on that curve in FIG. 4. Note that, in general, theboundaries between the segments S₁ through S₉ may fall anywhere alongthe trimming curve 11. In particular, they need not have any specialrelationship to the boundaries between the spans 12-15.

FIG. 5 depicts the piecewise representation of the segmented trimmingcurve 11 of FIG. 4. Note the use of different parametric equations foreach segment, and that each set of those equations is defined over itown associated span in t space 20.

During the discussion of FIG. 1 and the patch generation functions thenotion of polygons was briefly introduced. Beginning now with FIG. 6 wereturn to this topic, and examine in more detail the effects of trimmingupon the technique of rendering a surface with polygons.

FIG. 6 shows four spans 12-15 that map, according to the heavy arrows,into the four patches 16-19. Each of the spans is traversed by atrimming curve 11, whose purpose is to define the trimmed (or"subtracted") region 9 in the resulting surface. At present our interestin FIG. 6 is in the generation of (untrimmed) polygons; the effects oftrimming curve 11 upon those polygons will be considered beginning withthe Figures and text that follow FIG. 6.

The process of generating the patches is carried out one patch at atime. Accordingly, spans are evaluated one at a time; each span producesone associated patch. The evaluation of a span involves thedetermination of step sizes for u and v. These step sizes are notnecessarily the same for u and v, and each is again determined uponbeginning the evaluation of the next span. That is, the step sizes deltau_(i) and delta v_(i) for span 13 need not be equal to each other, norneed they bear any relationship to the step sizes delta u_(j) and deltav_(j) for span 12. In particular, there is no requirement that deltau_(i) equal delta u_(m) or equal delta u_(n), even though for simplicitythey appear to be equal in the figure. A similar statement applies tothe delta v's.

The user of the graphics system will have at least indirect control overthe step sizes. In some systems it may be possible for the user tospecify them directly, although it is more likely that the specificationis arrived at indirectly. For example, the user may instruct thegraphics system to choose step sizes such that the resulting polygonedges have approximately a particular number of pixels. Bear in mindthat, in general, the step sizes can be redetermined afresh for eachspan.

The selection of step sizes for a span determines a collection of (u,v)pairs within that span. This is represented by the dotted lines withingthe spans 12-15. The intersections of the dotted lines with themselves,and of the dotted lines with the solid lines representing the boundariesof the spans, are the points at which the patch generation functions areevaluated. For example, the evaluations of the patch generationfunctions at points 21-23 produces polygon vertices 24-26 (These pointsare simply arbitrary vertices in different polygons).

There are two further things to note before leaving FIG. 6. First, theadjoining edges of the polygons in the patches 16-19 are straight linesegments. An attempt has been made in the figure to show this. Incontrast, the patch boundaries and the edge of the trimmed region 9appear much smoother. They too are rendered with straight line segments,but with ones that are significantly shorter. That is, there are morepoints in uv space evaluated to produce those lines in the patches.Essentially, there is a mechansim for slipping extra points into thespan evaluation process. This notion of differential density will befurther discussed among the topics that follow. Second, the processdescribed above of evaluating patch generation functions at points in uvspace evaluates points in in all portions of the span, including thosethat will eventually be trimmed away. Trimming is a fairly complicatedprocess, requiring the analysis of several possible situations. Forexample, a given vertex may belong to a polygon that is totallyunaffected by trimming, one that is to be trimmed away in its entirety,or, to one that is crossed by the trimming curve (as it is mapped intoXYZ space). In this latter case part of the polygon remains and part ofit does not. This requires changing the shape of the polygon to matchthe ("mapped") trimming curve. That, in turn, will require the findingof new vertices for that polygon; ones that were not produced by theevaluation of the span at the steps in u and v as shown in FIG. 6.

Refer now to FIG. 7. What is shown there is an expansion of a portion ofFIG. 6. In particular, portions of patches 16 and 17 are depicted, alongwith a portion of the mapped trimming curve 27. By "mapped" trimmingcurve, we mean the trimming curve 11 as mapped into the patches by thepatch generation functions.

Two kinds of polygon vertices along patch boundaries are shown in FIG.7. The open (or hollow) circles correspond to the mapping into thepatches of: (a) the points of intersection of the dotted lines of FIG. 6with the span boundaries, and (b) the points of intersection of thedotted lines with themselves. The closed (or solid) circles correspondto "extra" vertices that are added by evaluating additional (u,v) pairsalong the span boundary. These extra points divide each step of delta uinto equal subportions; the steps of delta v are similarly divided. Itis common for the user to have some degree of control over the processof adding these extra vertices.

FIG. 7 also shows polygon vertices along the mapped trimming curve 27.It is clear from an examination of the figure that a good many extravertices have been added to the polygons as part of the trimmingprocess. We shall have much to say about this, and turn now to FIG. 8,which is a further expansion of that portion of the mapped trimmingcurve 27 pertaining to patch 17.

There are basically two topics that are of interest in connection withFIG. 8. The first is where the solid black squares come from. They arepoints along the mapped trimming curve 27 that are taken to be polygonvertices for polygons that are crossed by the mapped trimming curve 27.It will be recalled (see FIG. 5) that the trimming curve 11 is composedof segments. For the particular trimming curve 11 in FIG. 5 that servesas the basis of our example here in FIG. 8, segments S₆ through S₈ arethose that trim patch 17.

Segments S₆ through S₈ (as well as all the other segments of that or anyother trimming curve) are evaluated by finding, for each segment, adelta t that ideally produces steps in u and v that are each roughlyequal to the distance, in uv space, between the points along the spanboundary that correspond to an open circle and its nearest solid blackcircle neighbor on patch boundaries of FIG. 7. This business ofselecting the delta t's is influenced by the user's choices made inconnection with extra vertices for patch boundaries. In reality (and incontrast with the ideal case), the evaluation of the trimming curve 11,as described above, produces many (u,v) pairs that are spaced too closetogether to be useful. A further refinement is to examine the pairsproduced by the evaluation of the parametric functions (for the exampleat hand these are F_(u6) & F_(v6), F_(u7) & F_(v7), F_(u8) & F_(v8)) andsuppress pairs that are insufficiently distant from their predecessors.The solid black squares of FIG. 8 (and those of FIG. 7, too) result fromevaluating the patch generation functions at these refined (u,v) pairs.

The second thing that is of interest in FIG. 8 is the open squares.These represent the intersection of the mapped trimming curve 27 and thepolygon edges. Rather than find the intersections in XYZ space, thecorresponding point of intersection is found in uv space and then mappedinto XYZ space. Specifically, what is found is the intersection of twostraight line segments in uv space. One of the straight line segments isbetween two consecutive refined (u,v) pairs along the trimming curve 11.The other straight line segment would be a portion (e.g., 28 or 29 inFIG. 6) of the dotted lines or span boundaries of FIG. 6.

Before leaving FIG. 8, notice also the dotted lines that representpolygon portions and entire polygons that are determined to be invisibleby the trimming process, and so are not displayed.

We turn now to FIG. 9. That figure is a simplified pseudocodedescription of activity performed by a transform engine portion of ahigh performance graphics system hardware apparatus. That hardwareapparatus is the subject of description that appears in a later portionof this Specification and its Figures. The activity mentioned above isprimarily the trimming of polygons as illustrated in FIG. 8. FIG. 9 maybe understood as a condensed road map for accomplishing the type oftrimming described in connection with FIG. 8.

In a preferred method the graphics system is capable of displayingseveral B-spline surfaces, each of which may be variously trimmed. Eachsurface would generally be composed of a plurality of patches. Thetrimming activities occur at the level of polygon handling within eachpatch. Accordingly, steps 1 and 2 (in conjunction with steps 15 and 16)of FIG. 9 apply the process to be described in connection with steps3-14 to each of the patches in all of the surfaces. Step 2 isaccomplished in software executed by the computer associated with thegraphics system. Among other things, step 2 divides the surface intopatches, selects a patch to be rendered, computes u_(min), u_(max),v_(min), and v_(max) for the associated span of uv space, and decideswhich segments of which trimming curves will be needed for trimming thepatch. Steps 3-14 generate trimmed polygons vertices that can bedisplayed by a polygon rendering mechanism in the hardware of thegraphics system.

We now consider the activity within the range of the FOR loop of steps 3through 14. That activity includes the primary generation of the variouspolygons (as indicated by steps 3 and 14), and their subsequent (andimmediate) trimming (as indicated by steps 4 through 13).

The general order of polygon generation within a patch is illustrated inthe example span 30 of FIG. 10. That span is partitioned, in thisexample, into sixteen subspans; one for each untrimmed polygon to begenerated. The size of each subspan is determined in accordance with theuser's instructions pertaining to the desired visual smoothness of thesurface. In a preferred method the polygons are generated by a techniquecalled forward differencing, which does not require the explicitcalculation of the u and v values for the boundaries of each of thesubspans. Forward differencing is preferred because it is faster thanevaluating the patch generation functions at each of the subspancorners. For a description of forward differencing see Foley and VanDam, pages 535-536.

The trimming activities that are to follow require the use of cliplimits that correspond to the subspan for the polygon being generated.As usual, the term "clip limits" refers to a window of interest. Objectswithin the window are retained, while those that fall outside arediscarded. The clip limits that will be of interest to the trimmingdescribed herein are the subspan boundaries in uv space that areassociated with the generated polygons.

Unfortunately, as noted above the operation of computing polygonvertices in XYZ space through forward differencing does not explicitlyprovide the u and v values that are the subspan boundaries. If the cliplimits noted above are to be used, then they must be found separately.Step 4 computes the clip limits (in uv space) for the polygon currentlybeing generated and trimmed. FIG. 9A includes a subspan 35 having cliplimits u_(left), u_(right), v_(top) and v_(bottom). Subspan 35 could,for example, correspond to subspan #6 in FIG. 10: u_(left) would be thevalue 31, u_(right) would be the value 32, v_(top) would be the value34, and v_(bottom) would be the value 33.

Referring to FIG. 10, note that finding the clip limits for the varioussubspans requires determining the u values 31, 32, . . . and v values33, 34, . . . that partition the u and v axes. In a preferred embodimentthese are 32-bit floating point binary numbers. One can envision aprocedure for independently finding the four corners of any selectedsubspan. Any such procedure such be avoided unless it can be absolutelyguaranteed that the same (u,v) pair (i.e., their bit patterns areidentical) is always obtained for a shared corner (i.e., u value 31 andv value 33 describe a common corner for subspans #1, #2, #5 and #6 inFIG. 10), regardless of which subspan is currently of interest. Ratherthan make such guarantees for independently selected subspans, apreferred method, described next, allows the polygons to be generated atseparate times while ensuring, by saving certain principal values ofinterest, that common corners have identical u values and identical vvalues.

As shown in FIG. 10, the order of polygon generation is from from leftto right within rows, and then row-wise from bottom to top. Step sizevalues (delta u and delta v) are found as previously described. Whiletraversing a row (say, from subspan #2 to subspan #3) the u_(right) ofsubspan #2 is saved and reused as the u_(left) of subspan #3. At thebeginning of a row u_(min) for the span is taken (exactly) as u_(left).Along a row a new u_(right) is found by adding delta u to the old valueof u_(right). At the end of the row u_(max) for the span is taken(exactly) as u_(right). The values u_(min) and u_(max) arise when theB-spline generation software segments a surface into patches (see FIG.1).

Upon starting the first row v_(min) is taken (exactly) to be v_(bottom).The value for v_(top) is found by adding delta v to v_(bottom).Thereafter, at the start of a new row the old value of v_(top) is savedand reused as the new v_(bottom). The new v_(top) is found by addingdelta v to the old value of v_(top). When doing the very top row ofsubspans v_(max) is taken (exactly) to be v_(top).

Step 5 (in conjunction with step 10) of FIG. 9A may be better understoodwith reference to FIG. 11. That figure shows a span 36 crossed by anumber of associated trimming curves 37-40. Recall that trimming curvesare generally defined as segmented B-splines. When considering whatactions to take in trimming a particular patch, only those trimmingcurve segments that are inside the associated span need be considered.In accordance with what was said in connection with step 2 of FIG. 9, amechanism in the software of the graphics system will associate one ormore lists of trimming curve segments with each span. If, for example,the patch for the span 36 of FIG. 11 were to be rendered, then fourlists of trimming curve segments (i.e., their B-spline descriptions),the particular patch generation functions, the span definition, as wellas some other stuff, would be transmitted to the mechanism that executessteps 3-14 of FIG. 9. Of the four lists of trimming curves in ourexample, the two associated with trimming curves 37 and 38 will bemarked as describing "closed" trimming curves, while the two lists fortrimming curves 39 and 40 will be marked as "open". By this we mean thatthe lists for trimming curves 37 and 38 describe complete curves thatclose upon themselves. However, portions of trimming curves 39 and 40may be discarded as having no effect upon the trimming of the patchproduced by rendering span 36. (The discareded portions may, however, beneeded when rendering some other patch, and will appear in listsaccompanying the span for that other patch.) Open trimming curves muststart and end outside of their associated span. In a preferred methodthe number of trimming curve segments to be included in a list for anopen trimming curve will be increased to include starting and endingsegments that are "clearly" inside the span or "clearly" outside thespan, beyond any possible doubt or error that could be introduced byroundoff errors. This may be accomplished, for example, by consideringthe span to be, say, five percent bigger than it really is, as shown bythe dotted line 41.

We turn now to step 6 of FIG. 9, which (in conjunction with step 9)concerns the notion of straight line segments along a trimming curve.The task of trimming is going to involve finding many points along eachtrimming curve. Recall, for example, FIG. 8; all of the open and solidsquares along the mapped trimming curve 27 have to be found (locally, asit were) as a part of the trimming process. In particular, the "straightline segments" of step 6 of FIG. 9 correspond to straight lines betweenthe solid squares of FIG. 8. Implicit in step 6 is the finding of (u,v)pairs which, when evaluated by the patch generation functions 4, willproduce the solid squares. It is the finding of those (u,v) pairs thatis of interest at the moment.

It may be recalled that the mapped trimming curve 27 of FIG. 8corresponds to segments S₆ through S₈ of the trimming curve shown inFIG. 5. Recall also that the same trimming curve is shown in uv space inFIG. 6. The (u,v) pairs that we presently seek belong to the same uvspace as shown in FIG. 6. We last used FIG. 6 in connection with adiscussion of how polygons arose; here a similar operation is going toproduce a separate collection of (u,v) pairs to construct the straightline segment approximation of each B-spline segment of each trimmingcurve.

What is desired are (u,v) pairs along the trimming curve havingapproximately the same spacing therebetween as the extra vertices addedto the span boundaries for polygon generation. That is, the steps in ualong the trimming curve should be no larger than the distance in ubetween the extra vertices. If there are no extra vertices, then themaximum distances should be taken as the distance between cornervertices. A similar requirement applies to steps in v along the trimmingcurve. This collection of (u,v) pairs is a separate collection of pointsdifferent from any other, say, that used for polygon generation. Thesteps along the trimming curve will correspond to the solid squares ofFIG. 8. They will be used to determine additional (u,v) pairs thatcorrespond to the open squares of FIG. 8. We now describe how theinitial steps along the trimming curve are found.

Recall that the trimming curves are described by parametric equationshaving t as their independent variable. Knowing what step sizes in u andv are desired does not immediately imply a step size in t for actuallyevaluating the trimming curve functions. Furthermore, it would bedesirable if the technique of forward differences could be used toevaluate the trimming curve functions, just as was described for thepatch evaluation functions. That requires a uniform step size in t. So,what is needed is a step size in t that remains unchanged throughout theevaluation of the trimming curve segment, and that produces asufficiently dense collection of (u,v) pairs for use as the solidsquares of FIG. 8.

The basic idea is to evaluate each segment of each trimming curve atsome selected (and manageably small) number of points. The largest ofthe resulting increments in u and the largest of the resultingincrements in v are each saved. These increments are compared to therespective distances in u and v between the extra vertices along patchboundaries. If the comparison is favorable the desired step size in thas been found. If the comparison is unfavorable, then the step size int is adjusted in accordance with the ratio of the largest resulting stepsize and the desired step size. This determination process is performedseparately for u and v, resulting in two candidates for a step size int. The smaller of the two step sizes is selected.

Step 7 in FIG. 9 may be understood with reference to FIG. 12 and anappreciation of the Cohen & Sutherland clipping algorithm. (For adescription of that algorithm, see pages 146-149 in Foley and Van Dam.)FIG. 12A depicts a straight line segment 42 (having a direction that isfrom P₁ to P₂) that crosses a clip window 43 of interest. The purpose offinding a uniform step size in t and using the forward differencing toevaluate the trimming curve function (in the direction of increasingvalues of t) is to find a sufficient number of points along the trimmingcurve. These points are the solid squares of the figures, and thedirected straight line segments between them are a close approximationof the ideal trimming curve described by the B-spline functions.Clipping finds the points of intersection (if there are any) betweeneach staight line segment 42 and the clip window 43. This allows thedetermination of exactly what part 44 of the staight line segment 42lies within the clip window 43. In the example of FIG. 12A that part 44is the line between NEWP₁ and NEWP₂. NEWP₁ and NEWP₂ are computed by theclipping process. The subscripts indicate that the line segment 44 has adirection. As will become apparent, the trimming method to be describeduses the notion of directed motion along the trimming curves. Simplyput, as you travel along the trimming curve (or its straight lineapproximation) the stuff on the right gets trimmed away.

The Cohen & Sutherland clipping algorithm mentioned above is employed inconjunction with a number of enhancements. With reference now to FIG. 12and to the way the Cohen & Sutherland clipping algorithm is described atpages 146-149 in Foley and Van Dam, the preferred method incorporatesthe following improvements:

1. The clip limits are checked in the order u_(left), u_(right),v_(bottom), v_(top) because this is the order in which the polygons aregenerated (i.e. left to right within a row and row-wise from bottom totop).

2. Depending on the relative position of the clip window and thestraight line segment, either a NEWP₁ or a NEWP₂, or both, is computed.(FIG. 12 shows only two of many possible situations.) The computation ofsuch an intersection point at any stage of the clipping algorithm isalways done relative to the original P₁ to ensure that identicalintersection points are computed for adjacent clip windows, as shown inFIG. 12B. When considering window₁ and the straight line segment from P₁to P₂, the intersection point labeled NEWP₂ in window₁ will be computed.When considering the same straight line segment for window₂, theintersection point labeled NEWP₁ in window₂ will be computed. Thenumerical values computed for NEWP₁ must be identical to the numericalvalue computed for NEWP₂. The equations

    NEWV.sub.2 =v.sub.1 +((u.sub.right -u.sub.1)/(u.sub.2 -u.sub.1)) (v.sub.2 -v.sub.1)

    NEWV.sub.1 =v.sub.1 +((u.sub.left -u.sub.1)/(u.sub.2 -u.sub.1)) (v.sub.2 -v.sub.1)

where:

u₁ and v₁ are the coordinates of P₁

u₂ and v₂ are the coordinates of P₂

give the value of the v coordinate of the intersection point for the twowindows. The u coordinate is u_(right) for window₁ and u_(left) forwindow₂.

3. After clipping against the left, right, bottom, and top boundaries itis possible (because of floating point roundoff errors) for onecoordinate of either NEWP₁ or NEWP₂ to be slightly outside of the clipwindow. This is corrected by moving that outside coordinate of NEWP₁(and/or NEWP₂) to the clip boundary after clipping. This check is madefor all four clip boundaries. (Roundoff errors could just as easily moveone coordinate of the final points to the inside of the clip window.This case does not cause problems later in the trimming activity, andmay be ignored.)

4. If the trimming curves are represented as B-splines, it willtypically be the case that the (u,v) pair obtained when a given segmentis evaluated at the end of its span (i.e., for the largest value of t inthat span) will not exactly equal the (u,v) pair obtained when the nextsegment is evaluated at the beginning of its span (i.e., for thesmallest value of t in the other span). This may be the result offloating point roundoff errors or of the failure of the B-splineapproximations to converge exactly at their "joint" between segments.This problem is avoided by using the first point of the later segment inplace of the last point of the former segment. Thus, the last straightline segment of the straight line segment approximation for a giventrimming curve segment goes from the next to last point of that segmentto the first point of the next segment.

5. Typically, P₂ of a given straight line segment becomes P₁ of the nextstraight line segment. When this occurs, the old value of P₂ can becopied into P₁. Then the clipper can save information about the positionof P₂ relative to the clip window and avoid recomputing thisinformation.

6. The clipper sets three flags (which will be used later):

ACCEPT :=TRUE if the straight line segment from P₁ to P₂ is partially orcompletely inside the clip window;

CNEWP1 :=TRUE if a new P₁ was computed (i.e., P₁ was outside the clipwindow); and

CNEWP2 :=TRUE if a new P₂ was computed (i.e., P₂ was outside the clipwindow).

7. If the ACCEPT flag is TRUE the clipper updates NEWP₁ and NEWP₂. If P₁is on or inside the clip window then NEWP₁ =P₁ ·P₂ is handled similarly.

Refer now to step 8 of FIG. 9. After each straight line segment isclipped against the current clip window, information about points NEWP₁and NEWP₂ is saved in a list of tables if the ACCEPT flag is TRUE. Ingeneral, a trimming curve can have many points within the current clipwindow. (FIG. 8 is helpful here; it shows polygon vertices along themapped trimming curve 27. In each subspan the clipping process returneda plurality of points for the trimming curve.) A trimming curve mayenter a clip window and continue within it for many points beforeexiting. It may exit and then reenter. It might even simply start (fort=0) some place in the middle of the clip window. A trimming curve mightintersect a clip window at exactly one corner of the window. In terms oftables added to the list then, a given trimming curve may produce one ormore tables to be added to the list. Because trimming curves have adirection associated with them (increasing values of t), it is useful todistinguish among: (a) points of intersection along the boundary of aclip window where a trimming curve enters the window; (b) points along atrimming curve that are inside the clip window; and (c) points ofintersection along the boundary of a clip window where a trimming curveleaves the window. These points will be called "BEGIN", "MIDDLE", and"END" points, respectively. In general a clip window can be cut byseveral trimming curves. It is desirable to group the tables for eachindividual trimming curve into separate circularly linked sublistswithin the larger list of all of the tables generated for a particularclip window.

The general format of the tables is shown in FIG. 9. The first twoentries 45 and 46 in a table are the binary floating point values of theu and v coordinates for the point of interest (either NEWP₁ or NEWP₂).The third entry 47 contains three parts: either zero or a nonzerotrimming curve number (an 8-bit integer); either zero or a nonzeroB-spline segment number (also an 8-bit integer) identifying the trimmingcurve on which the point in uv space lies; and six flag fields. (Zerosin the first two parts indicate special cases described later in a moredetailed description of a preferred implementation.) The six flag fieldsare:

FF1 A two bit field indicating whether this table is for a "BEGIN","MIDDLE", "END", or "OTHER" point. "OTHER" points include corners andextra vertices, as will be described below.

FF2 A two bit field indicating which edge of the clip window (i.e.,left, right, bottom, or top) the point lies upon. This field is usedmainly for tables describing corners and extra vertices.

FF3 A one bit field indicating if this table should be skipped. This bitwill be set if this table is one of the special cases to be described inconnection with step 11 of FIG. 9. This flag is called the SKIP flag.

FF4 A one bit field which will be used in step 12 of FIG. 9 to indicatethat this table has been handled already. This flag is called the MARKflag, and is initially FALSE.

FF5 A one bit field which will be used in step 12 of FIG. 9 to indicatethat this table is the first table to be handled. This flag is calledthe START flag, and is initially FALSE.

FF6 A one bit field which will be determined in step 11 of FIG. 9 andwhich indicates if the vertex in XYZ space corresponding to this tableremains present "ON" the surface (it has not been trimmed away) or isabsent from, or "OFF", the surface (it has been trimmed away). This flagis called the ON flag.

(Remaining present ON the surface does NOT mean the vertex will actuallybe visible. Hidden surface removal, back cull and clipping operationscan prevent a point that is present ON the surface from ultimately beingdisplayed. On the other hand, a point that is absent OFF the surface isNEVER visible.)

We shall have many future occasions to refer to the presence or absenceof a vertex, or to whether it is ON or OFF the surface. In the same wayit will be convenient to refer to polygons and entire patches as eitherpresent or absent, ON or OFF. Polygons and patches do not haveindividual ON flags of their own; their presence or absence will bedetermined by the presence or absence of certain vertices within them.The ON flags of those important vertices are saved as they areencountered.

The fourth entry 48 contains the Wrap Address and the Next Address.Nonzero Wrap Addresses are the circular links in an otherwise linear(i.e., physically ordered) list of tables. A Wrap Address of zeroindicates that the next physical table is also the next logical table inthe list. Tables are inserted onto the end of the list of tables in anorderly manner as trimming curves are traversed in the direction ofincreasing values of t. To make the desired circularly linked sublistfor a trimming curve it is merely necessary to insert the address of thefirst table for that trimming curve into the Wrap Address of the lasttable for that trimming curve.

The Next Address is determined by steps 11 and 12 in FIG. 9. It linksthe tables into another list used to display portions of polygons thatare present.

As part of the discussion of step 8 of FIG. 9, it is appropriate todescribe the criteria used to build the list of tables in response toinformation provided by the clipper. In general, depending on theposition of the line from P₁ to P₂ relative to the clip window, it willbe necessary to add zero, one, or two tables to the list. The followingdecision tables describe how this is done. In these decision tables theflag INSIDE is used to remember if P₂ is inside or outside the clipwindow.

    ______________________________________                                        DECISION TABLE FOR THE FIRST                                                  (P.sub.1,P.sub.2) PAIR ALONG EACH TRIMMING CURVE                              If ACCEPT = FALSE then INSIDE := FALSE                                        If ACCEPT = TRUE then                                                         CNEWP1  CNEWP2                                                                F       F         If the current trimming curve is an                                           open loop then add to the list a                                              BEGIN table for NEWP.sub.1, else do                                           nothing.                                                                      If P.sub.2 is the last point on an open                                       loop trimming curve then add to the                                           list an END table for NEWP.sub.2, else                                        add a MIDDLE table for NEWP.sub.2.                                            INSIDE := TRUE.                                             F       T         If the current trimming curve is an                                           open loop then add to the list a                                              BEGIN table for NEWP.sub.1, else do                                           nothing.                                                                      Add to the list an END table for                                              NEWP.sub.2.                                                                   INSIDE := FALSE.                                            T       F         Add to the list a BEGIN table                                                 for NEWP.sub.1.                                                               Add to the list a MIDDLE table for                                            NEWP.sub.2.                                                                   INSIDE := TRUE.                                             T       T         If NEWP.sub.1 < >NEWP.sub.2 then add to the                                   list a BEGIN table for NEWP.sub.1                                             and an END table for NEWP.sub.2, else do                                      nothing.                                                                      INSIDE := FALSE.                                            ______________________________________                                    

    ______________________________________                                        DECISION TABLE FOR THE REMAINING                                              (P.sub.1,P.sub.2) PAIRS OF EACH TRIMMING CURVE                                If INSIDE = TRUE then                                                         CNEWP1  CNEWP2                                                                F       F          If P.sub.2 is the last point on an open                                       loop trimming curve then add to the                                           list an END table for NEWP.sub.2, else                                        add a MIDDLE table for NEWP.sub.2.                         F       T          Add to the list an END table for                                              NEWP.sub.2.                                                                   INSIDE := FALSE.                                           T       T / F      Can't Happen.                                              If INSIDE = FALSE then                                                        If ACCEPT = FALSE then go to next (P.sub.1, P.sub.2) pair                     If ACCEPT = TRUE then                                                         CNEWP1  CNEWP2                                                                F       T / F     Can't Happen.                                               T       F         If P.sub.2 is the last point on an open                                       loop trimming curve then add to the                                           list an END table for NEWP.sub.2, else                                        add a MIDDLE table for NEWP.sub.2.                                            INSIDE :=  TRUE.                                            T       T         If NEWP.sub.1 < >NEWP.sub.2 then add to the                                   list a BEGIN table for NEWP.sub.1                                             and an END table for NEWP.sub.2, else do                                      nothing.                                                    ______________________________________                                    

Steps 9 and 10 cooperate with steps 6 and 5, respectively, describedabove. The result of these nested FOR loops is the division of eachtimming curve into straight line segments. Those are then clipped bystep 7 against the subspan whose polygon is to be rendered. Step 8builds a list of tables describing the clipped straight line segments.

In steps 11(a)-(d) the list of tables produced at step 8 is processed infour ways. The first of these is an inspection by step 11(a) for the twospecial cases shown in FIGS. 3A-B.

Referring now to FIG. 13A, what is shown there is a subspan 49 and atrimming curve 50 having a point 51 that falls "just outside" the clipwindow. According to the clipping algorithm described in step 7 of FIG.9, a point of intersection 52 will be calculated for where the trimmingcurve 50 "goes out" of the clip window 49. According to the example, thetrimming curve comes right back in again. The clipper will dutifullyproduce another point of intersection 53. If the point 51 issufficiently close to the clip limit (which in the example is theright-hand edge, but it could be any edge) the finite resolution of thenumbers used in the calculations can result in both the exit andentrance points lying on top of one another. This is what FIG. 13A isall about; within the degree of arithmetic precision in use, points 52and 53 are the same point. This is really what we mean by "just barely"outside.

When the situation described above occurs, two tables having certainproperties will be consecutively added to the list. In particular, thefirst will be an END table (see FF1 in connection with step 8 of FIG. 9)and the second will be a BEGIN table. Furthermore, both tables will haveexactly identical u values and exactly identical v values. This state ofaffairs is the first special case associated with the processing of thelist of tables by step 11(a).

The first special case is found by inspecting each of the circularlylinked sublists associated with the trimming curves. If any sublistcontains logically adjacent (i.e., with the Wrap Address taken intoproper account to produce correct logical ordering) BEGIN and END tableseach containing the same identical (u,v) pair, then each of those tablesis converted to a MIDDLE table.

The second special case that is checked for is depicted in FIG. 13B.This case is generally analogous to the first special case, except thatthe trimming curve 54 extends "just barely" into the clip window 55. Inparticular, the finite resolution of the numbers used in the calculationof points of intersection will produce a BEGIN table and an END tablefor points 56 and 57 having the same (u,v) pair. However, in this casethe BEGIN and END tables will be separated by a MIDDLE table for point58.

The second special case is found in step 11(a) by an inspection of thelist for sublists containing BEGIN--MIDDLE--END sequences having BEGINand END tables containing the same identical (u,v) pair. When such asequence is found the SKIP flag (refer to step 8 of FIG. 9) is set inall three tables. The net effect of this upon later processing of thosetables is their effective deletion; they are simply ignored.

In step 11(b) the list of tables is processed to make four new sublists;one sublist for each edge of the clip window, with each such sublistenumerating the trimming curves which cross the associated edge. Thesenew edge sublists are formed in place, as it were, by using the NEXTADDRESS field of the tables to link the edge sublists together. (Thiswill be only an intermediate use of the NEXT ADDRESS field, not itsfinal use.)

Only BEGIN and END tables without their SKIP flags set are included inthe edge sublists. In forming the edge sublists special attention willbe paid to cases where the (u,v) pair in a BEGIN or END table is also acorner point of the clip window. In this connection it will be helpfulto refer now to FIG. 14, which depicts a clip window crossed by severaltrimming curves. A BEGIN or END table whose (u,v) pair also (by chance)describes the corner point I goes into the edge sublist for the top edge59. Similarly, a BEGIN or END table at corner II goes with the left edge60. A BEGIN or END table at corner III goes with the bottom edge 61. Inlike fashion, a BEGIN or END table at corner IV goes with the right edge62.

Once formed, each edge sublist is sorted by either ascending ordescending values of either u or v, depending upon which edge thesublist represents. The attributes BEGIN and END do not influence thesorting. The edge sublist for the left edge is sorted according todecreasing values of v. The edge sublist for the bottom edge is sortedaccording to increasing values of u. The edge sublist for the right edgeis sorted according to increasing values of v. The edge sublist for thetop edge is sorted according to decreasing values of u.

At step 11(c) the list of tables is processed to interleave the fouredge sublists with tables in the main list for corners and extravertices to produce a new circular ordering in the list. This is thefinal use of the NEXT ADDRESS field. The interleaving begins(arbitrarily) with corner vertex I, and proceeds counterclockwise aroundthe boundary of the clip window. The NEXT ADDRESS field of each tablefor a point along the boundary is linked to the next point further alongthe boundary, whether that be an extra vertex, or an intersection with atrimming curve. It could even be the case that the next vertex is thenext corner. This reordering is represented in FIG. 14 by the arrowsconnecting the points labeled as C's, X's, B's and E's. Beginningarbitrarily with vertex I, the resulting ordering isC--X--E--X--X--C--X--B--X--E--C--B--C--E--B--C. The only types of tablesincluded in this reordering with the NEXT ADDRESS field are corners,extra vertices, and the BEGIN and END intersection point tables; leftundisturbed are the sublists for each trimming curve enumerating thepoints along each trimming curve within the clip window (the MIDDLE'sbetween the BEGIN's and END's).

It might be observed at this point that nothing particular has been saidabout the way that tables for corner vertices and tables for extravertices are kept in the overall list of tables for the clip window ofinterest. Clearly, some mechanism is needed for adding and later findingsuch tables, so that the interleaving described above can beaccomplished. Various techniques are well known for accomplishing thissort of thing, and the one used in any particular embodiment willdepend, in large part, upon the inclinations of the designer. In thepresent embodiment they are located in their own linked sublistsidentified by associated pointers.

Step 11(d) of FIG. 9 calculates the ON flag for each table in thecircularly linked list just described above in connection with FIG. 14and step 11(c) of FIG. 9. The ON flag indicates if a vertex remainspresent ON the surface or is trimmed away. What interests us here is howthe ON flag may be calculated. Simply put, this is done by finding apoint where a trimming curve leaves the subspan. By definition, everypoint on a trimming curve is assumed to be present. Trimming involvesthe further assumption that everything in a subspan that is to the leftof the trimming curve (while proceeding along the trimming curve in thedirection of increasing t) is also present. Accordingly, if we start ata point on the subspan boundary that is also the exit of a trimmmingcurve, the next vertex in the counterclockwise direction, and allsubsequent vertices until a trimming curve is encountered, are alsopresent. The reader may satisfy himself as to the relationship between"to the left of (with increasing t)" and "counterclockwise around thesubspan" by an appeal to FIG. 14. Pick a vertex that is present ON thesurface and try going both clockwise and counterclockwise.

ON flags are calculated by picking any END table as a starting point,say, the point 63 in FIG. 14. Its ON flag already designates ON. Thenthe circular list of tables is traversed, and each ON flag is also setto designate ON, until a BEGIN table is encountered. The ON flags forsubsequent tables are cleared to designate OFF until an END table isencountered. This process is continued until the initial END point 63 isagain encountered. If this is carried out on the example shown in FIG.14, beginning with the END point 63, the following sequence of ON flagswill be obtained (1/0=ON/OFF): 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0,0.

It might be the case that the circularly linked list for a subspancontains no BEGIN or END tables. This can happen in two different ways.First the subspan might have no trimming curves associated with it atall. Such a subspan will have only tables for corners and extravertices. In the second case a subspan might wholly contain one or moretrimming curves. (These might touch or coincide with the subspanboundaries, but not cross them, and the effect will be the same when thecurve is entirely inside without touching.) Such a subspan will giverise to a list having tables for corners, extra vertices and MIDDLE'sfor the trimming curve(s).

When rendering a polygon corresponding to the first case above thepresence of that polygon will be determined by the presence of adjoiningpolygons; it will be either totally present or totally absent. In thesecond case the trimming curve is smaller than the smallest "unit ofinterest" (i.e., the polygon). The trimming curve will be ignored andthe entire polygon tentatively retained. However, it does notnecessarily follow that the polygon will remain present; it might itselfbe within a larger trimming curve, and so be trimmed away. In the secondcase, as in the first, the presence of the polygon will depend upon thepresence of adjoining polygons.

The second case above remains as described even when the interiortrimming curve itself contains another trimming curve. Presumably thissituation corresponds to alternating regions of presence and absenceupon the surface. However, such features are too small to be seen withinthe chosen polygon size. Do not confuse that situation with the casewhen polygon size is decreased to where the trimming curves begin tocross subspan boundaries. That simply produces normal trimming, asusual.

It is clear from the foregoing that during the processing of eachindividual polygon some mechanism is needed to remember whether certainpolygons were present or absent. The mechanism will involve saving thevalues of the ON flag for certain of the tables for selected subspansduring the processing of a patch. Referring now to FIGS. 10 and 14,recall that subspans are taken in a definite order. For the firstsubspan within each row the ON flag for vertex IV 64 is saved for use inprocessing the next row. Within a row the ON flag for vertex III of eachsubspan is temporarily saved for use in rendering the next polygon inthat row. In addition, the ON flag for vertex III 66 of the last subspanof the first row is saved for use in processing a patch to the right ofthe current patch. (Although we haven't said so to this point, a surfaceis made up of row and columns of patches; FIG. 10 could refer to that aseasily as to subspans within a patch.) Similarly, (to accommodate rowsof patches) the ON flag for vertex I 67 of the first subspan of the lastrow of subspans within a patch is also saved, after being complementedif there is a BEGIN or END table exactly coincident with vertex I. Suchsaving is also for use in processing a subsequent patch; thecomplementing accounts for a special case explained in the paragraphsthat follow.

Recall that in connection with step 11(b) of FIG. 9 it was said that aBEGIN or END table that by chance had the same (u,v) pair as vertex Iwould be included in the edge sublist for the top edge of the subspan.At step 11(c) it was said that the four edge sublists were interleavedcounterclockwise around the subspan, beginning with vertex I. This meansthat in the special case where a trimming curve crosses a boundary bypassing through vertex I 67 there are in the circularly linked listactually two tables describing vertex I; one for vertex I itself and onefor the BEGIN or END coincident with vertex I. Each of those tables hasan ON flag; special attention must be paid to how to interpret these inlight of the rules by which they were set and the special case at hand.

There are three previously mentioned rules or conventions that apply tothe ON flag. These are:

1. All points along a trimming curve are present ON the surface.

2. Proceeding along a trimming curve in the direction increasing valuesof t, points of the surface to the left of the trimming curve arepresent ON the surface.

3. Proceeding counterclockwise around the subspan, vertices immediatelysubsequent to an END have tables marked ON until the next BEGIN, whichwill also be marked ON. Similarly, vertices immediately subsequent to aBEGIN are OFF until the next END, which is itself marked ON.

Now suppose that a BEGIN is coincident with vertex I. According to rule3, vertex I will be marked OFF (absent). However, the adjacent patchabove the current patch is, by rule 2, ON (present). Hence, the value ofthe ON flag for vertex I 67 propagated to the next patch needs to bechanged from OFF to ON.

Suppose that an END is coincident with vertex I. According to rule 3,vertex I will be marked ON. However, the adjacent patch above thecurrent patch is, by rule 2, OFF. Hence, the value of the ON flage forvertex I 67 propagated to the next patch needs to be changed from ON toOFF.

Thus, the general rule for propagating the ON flag for vertex I 67 ofthe first subspan of the last row is: if there is in no BEGIN or ENDcoincident with vertex I 67, propagate the unchanged value of the ONflag for vertex I 67. If there is a BEGIN or END coincident with vertexI 67, propagate the complement of the value of the ON flag of vertex I67.

A specific rule is needed to calculate the ON flag for the very firstvertex of an entire surface. Such a rule amounts to a convention, butshould be consistent with the trimming operations that might apply tothat very first vertex. For example, assuming that the very first vertexis always present ON the surface might not allow that vertex to betrimmed away. A convenient rule for the calculation of the ON flag ofthe very first vertex is to require that surface edges or portionsthereof that are present (ON) are bounded by a trimming curve. With thisapproach the very first vertex can be assumed to be OFF. This assumedvalue is ignored if a trimming curve produces at least one BEGIN/ENDpair of vertex tables for the first polygon, since in such a case it ispossible, as shown above, to compute the correct value of the ON flagfor each vertex. In the absence of such a pair of vertex tables then theentire first polygon has indeed been trimmed away.

An interesting special cases arises when the trimming curve around theedge of the surface is the only trimming curve associated with the veryfirst polygon. In that case the left and bottom edges of that polygonare coincident with that trimming curve, and two sets of vertices arisedescribing those two edges. One set says that the corners and extravertices for those polygon edges are absent from the surface, while thevertices calculated for intersection of those edges with the trimmingcurve are, by definition, present ON the surface. By virtue of the waythe clipper works, the OFF corner vertices are replicated as vertices onthe trimming curve, thus preserving the shape of the first polygon. Themechanism that processes the linked list of vertex tables, describedbelow in connection with FIGS. 16 and 17, tolerates the duplicatedescriptions, and will ignore the OFF edges and produce a polygon fromthe ON edges. It won't care that two of the OFF edges were the same astwo of the ON edges. Thus is it that an OFF corner is present on an ONpolygon.

Now consider step 12 of FIG. 9. Its task is to traverse the list oftables. Step 11 added to the list information about what originalvertices remain present after trimming, as well as what trimming curvevertices are to be incorporated into the trimmed polygon. Note that step11 may have effectively divided the original polygon into two or moresmaller polygons. Step 12 uses a simple procedure of nested loops totraverse the data structure prepared by step 11. A simplifiedrepresentation of how that data structure would represent the situationof FIG. 14 is depicted in FIG. 15. The result of the traverse by step 12is sequences of present polygon vertices that are sent by step 13 to amechanism for actually displaying the polygons. Step 12 will produce onecomplete list of vertices for each present polygon or present subpolygon(a part of a polygon remaining after another part has been trimmed away)emerging from the trimming process.

FIG. 15 is a simplified illustration of one possible circularly linkedlist of vertex tables that could result from applying step 11C of FIG.9B to the example of FIG. 14. The assumption was made that trimmingcurve #1 (for this particular polygon) starts (t=0) at point 68 of FIG.14, and that the other two trimming curves start somewhere outside thesubspan. The entire circularly linked list of FIG. 15 may be termed aboundary list. The Next Address field links untrimmed polygon verticesand points of intersection between the polygon edges and the trimmingcurves into the boundary list. Embedded within the boundary list aretrimming curve sublists. The trimming curve sublists are linked by acombination of physical ordering and the Wrap Address field. There isone trimming curve sublist for each trimming curve. When a BEGIN isencountered while traversing the boundary list, a choice can be made toeither continue along the boundary list or to switch over to traversingthe trimming curve sublist. (The boundary list and the trimming curvesublist share BEGIN and END tables; to switch all that is needed is tofollow the other chain of links.) If a trimming curve sublist istraversed to its END, the boundary will be reentered at that point.

Step 12 is depicted in FIG. 16 as a simplified flowchart, and in FIG. 17as pseudocode for operating upon the list of tables containing theinformation set out in step 8 of FIG. 9.

The flowchart of FIG. 16 is believed to be fairly self-explanatory whentaken in the context of what has preceded it. It includes an outer loop,shown in FIG. 16A, that selects an arbitrary vertex in the circularlylinked boundary list as an initial vertex that it then marks as havingbeen "first". The outer loop then traverses the boundary list. When itdiscovers that it has gotten back to the vertex marked "first", theprocess is complete, and the loop terminates. The main activity of theouter loop is to identify vertices that have not yet been visited andthat are ON the surface. When such a vertex is found it location in theboundary list is saved, so that the outer loop can resume traversing thelist at this point after the middle and inner loops have been allowed totraverse the list on their own. With the outer loop thus "suspended", asit were, further processing of the list by the middle loop commences.

The middle loop adds to a polygon list any present non-timming curvevertices and identifies a BEGIN or END vertex. When a BEGIN vertex isfound it is added to the polygon list and the inner loop is thenentered. An END vertex is also added to the polygon list because it is anext vertex present ON the polygon, and because it has now been markedas visited, and will not be further processed if it is visited again.

The inner loop traverses a trimming curve sublist whose head has beenreached by traversing along the boundary list with the outer and middleloops. The middle loop enters the inner loop after adding a BEGIN vertexto the polygon list; the inner loop returns to the middle loop when thebalance of the trimming curve sublist has been entirely traversed andthe associated vertices added to the polygon list. The middle loop thenresumes its traverse of the boundary list. The inner loop is re-used asneeded each time the head of a new trimming curve sublist isencountered. The middle loop will complete itself when an entire closedpolygon has been added to the polygon list. This condition is detectedwhen the middle loop returns to a vertex that has already been visited.This does not necessarily mean that the boundary list has been fullyprocessed; trimming can cut an original polygon into two or moresubpolygons. As long as the outer loop has not yet run to completionthere may be more subpolygons described by the boundary list.

Upon the addition of a complete polygon to the polygon list the middleloop returns to the outer loop. It resumes its processing at the pointin the boundary list where it left off. It continues to traverse thelist looking for vertices that have not yet been visited. If it findsone it re-uses the middle loop to put another entire subpolygon onto thepolygon list.

It is substantially easier to add vertex information to the polygon listfor corner and extra vertices (operation 69 in FIG. 16) than it is forpoints along trimming curves (operations 70 and 71 in FIG. 16). Theformer information is computed, as mentioned previously, by the forwarddifferencing technique. The latter information involves evaluating, forabritrary points in uv space, the vertex position in XYZ space and thesurface normal vector at the vertex. While not conceptually difficult,this is a much more time consuming operation.

FIG. 17 is a simplified section of pseudocode to perform the sameactivity as set out in flowchart form in FIG. 16. The difference is thatFIG. 17 incorporates more of the table structure described in connectionwith steps 8-11 of FIG. 9.

We conclude our exposition of a preferred method of trimming surfacepolygons with an examination of certain enhancements that may beincorporated into the method. We shall consider in the next fewparagraphs ways to minimize the effects of a less than idealparameterization of the trimming curve functions and to reduce to aminimum the number of trimming curves that must be considered whileprocessing each subspan and its polygon.

Ideally, one would like equal steps in t to produce equal increments inu and equal increments in v. This property arises from the nature of theB-spline functions chosen to represent the trimming curves, and mayobtain in greater or lesser degree. If such equality obtains it is saidthat there is a good parameterization for the trimming curve functions.A bad parameteization cannot always be avoided, and if left uncorrectedfor causes certain problems for the trimming process. Specifically, itcan cause the number of polygon vertices for trimmed polygons to becomeunduly large. That, in turn, increases the amount of memory that must beallocated for the trimming of polygons. Rather than add memory toaccommodate a phenomenon that has no particular benefit, a moresatisfying approach is to examine the uv pairs produced by evaluatingthe trimming curve functions with a selected step size in t, andsuppress those that are insufficiently distant from their predecessors.For further introduction to what we are about to discuss, refer to thedescription of FIG. 8.

Consider the trimming curve 72 shown in a span of uv space by FIG. 18.Equal step sizes of delta t 73 produce varying delta u's 74 and varyingdelta v's 75. Notice how the tick marks on the curve 72 are of varyingdistance apart; they are the evaluated coordinates in uv spacecorresponding to the equally spaced tick marks in t space.

From the discussion of FIG. 8 and of step 6 of FIG. 9 it will berecalled that a step size in t (delta t) is selected to produce maximumsfor delta u and delta v that are about equal to the distance betweenpolygon vertices along the edge of the patch, which distance may betermed the edge step size. That is how the delta t 73 of FIG. 18 isselected. For the purpose of illustration, assume that the largest deltav 75 shown in FIG. 18 was used to determine delta t. In the illustrationat hand the delta v's tend to be bigger than the delta u's. For adifferent trimming curve it might be the case that the largest delta uexceeds the largest delta v and would be used to select delta t. Theprocess of selecting delta t is described in connection with step 6 ofFIG. 9.

The straight line segment approximation of the trimming curve 72 that isobtained by evaluation at the equal step sizes of delta t 73 is refinedby requiring that at least one of delta u and delta v exceed fifty-twopercent of the edge mesh size. If neither does then that point in uvspace is ignored, and the next one is considered, and so on. Accordingto this technique, point 76 would be ignored, and straight line segment77 would run from point 78 to point 79. The first point along eachB-spline trimming curve segment is the initial point at which the pointselection process begins. This ensures that the B-spline trimming curvesegments join each other as intended.

The fifty-two percent criterion arises from a relationship that existsbetween the shortest and longest accepted straight line segments Therelationship involves an equality between the amount that a longestaccepted straight line segment (one at forty-five degrees) exceeds theedge step size on the one hand, and the amount the shortest acceptedstraight line segment (one that is horizontal or vertical) is less thanthe edge step size on the other hand. It may be shown that the desiredequality relationship obtains when the selection criterion isapproximately fifty-two percent of the edge step size.

The number of trimming curves that must be considered while processing asubspan and its polygon may be reduced by use of the followingtechnique. Simply put, information about the location of each trimmingcurve segment within the span is associated with its B-splinedefinition. This information is used to eliminate from considerationthose segments that cannot possibly intersect the subspan about to beprocessed.

Prior to the generation of the first polygon for a new patch av_(extent) is found for each B-spline segment of each trimming curve.The v_(extent) is the range in v from the minimum v value to the maximumv value in the set of (u,v) pairs resulting from the parameterizationrefinement described in connection with FIG. 18. The u_(extent)mentioned below is defined similarly. The v_(extent) for each B-splinesegment is stored in a location within a data structure containing thetrimming curves for the patch.

Recall that polygons are generated with a row-wise ordering of thesubspans. See the discussion of FIG. 10. At the beginning of each row ofpolygons the v_(extent) of each trimming curve segment for the patch isinspected. Those trimming curve segments having a v_(extent) thatoverlaps the range of v for the current row are further evaluated todiscover their associated u_(extent). The u_(extent) for each suchoverlapping sement is also stored in the trimming curve data structure.

For each polygon along the row only those B-spline segments whoseu_(extent) and v_(extent) both overlap the current subspan areconsidered in the clipping and trimming process described in connectionwith FIG. 6-16. For example, and with reference again to FIG. 6,trimming curve segment 80 would not be of interest for anysubspan/polygon in row 82 of patch 12; segment 80 does not have av_(extent) that overlaps the v_(extent) of row 82. On the other hand,the v_(extent) of segment 81 does overlap that of row 82, and theu_(extent) of segment 81 is computed prior to processing any of thesupspans along row 82. Because the u_(extent) of segment 81 overlaps theu_(extent) of the subspan 83, the B-spline segment 81 of trimming curve11 is included in the clipping and trimmng activities for that subspan.Segment 80 would not be included in the activities for anysubspan/polygon along row 82, nor would it, for further example, be usedin connection with subspan 84. And in the same manner, neither segment80 nor segment 81 would be included in the activities for anysubspan/polygon along row 85; no v_(extent) for those segments overlapsthat for the row.

Refer now to FIG. 19, wherein is shown a pictorial representation of anactual graphics system embodying the method of the invention. Inparticular, the graphics system includes a computer 86, a keyboard 87, aknob box 88, a button box 89 and a mouse 90. The computer 86 may executethe software, mentioned at the beginning of this Specification, whichinteracts with the user and prepares the B-spline descriptions of thesurface and its trimming curves. The computer 86 is coupled to a grahicsaccelerator 91 through a high speed local graphics bus 92. The graphicsaccelerator 91 is in turn coupled to a color monitor 94 through threecoaxial cables for carrying the Red, Green and Blue (RGB) video signals.

The trimming activities described above are performed entirely withinthe graphics accelerator 91. While it is conceivable that the trimmingmethod that has been described could be carried out in software in thecomputer 86, such an approach necessarily incurs a substantial penaltythrough reduced performance compared with trimming by appropriatehardware. To accomplish this and other goals, the graphics accelerator91 is responsive to the computer 86 through the execution of a number ofcommands. Among these are some that: instruct the graphics acceleratorto store in a data structure the definition of one or more trimmingcurves; store in a data structure a surface patch definition and thenrender it with trimming according to the previously defined trimmingcurves; and a command to delete all previously defined trimming curves.Also included are commands to push and pop a stack of ON flags. The ONflags for the upper right and lower left corners of selected patches aresaved on two separate stacks to facilitate the subdivision of patches.The subdivision of patches into subpatches arises when a given patchcontains too many trimming curves; there are practical and cost relatedconsiderations (i.e., the amount of memory) that limit the number oftrimming curves for any single patch that can be handled by the graphicsaccelerator 91.

The memory limitations mentioned above do not impose any practicalrestrictions on the number of trimming curves that may apply to a patch.The software in the computer determines the need for patch subdivisionwhen the number of trimming curve segments for a given patch exceedsapproximately forty to sixty. (The software can either anticipatelimitations in the graphics accelerator, or subdivide in response to anindicated overflow returned by the graphics accelerator ) A previouslyundivided patch can be divided up to sixteen times. Each act ofsubdivision can divide the patch at hand into four subpatches.

Each time a subdivision occurs the saved ON flags for the appropriateother patches already rendered are pushed on their stacks. There is onestack for vertex I of the first polygon of the last row of polygons ofeach patch, and another stack for vertex III of the last polygon of thefirst row of polygons of each patch. These stacks are separatelymaintained in cooperation with the software executed by the computer 86.The resulting mechanism for patch subdivision is transparent to thedescribed method for trimming patches. That method always operates thesame way, regardless of whether the patch at hand is a subpatch or not.

FIGS. 20A-C comprise a block diagram of the graphics accelerator 91shown in FIG. 19. An interface mechanism 95 couples the trimming curveand patch definition commands from the computer 86 to a transform engine96. The definitions are stored by a microsequencer 97 in a data RAM 98.The microsequencer operates upon these definitions to produce thecoordinates of the vertices for the trimmed polygons. Screen coordinatesfor the vertices of the trimmed polygons are sent via a DC (DeviceCoordinate) Bus 99 to a scan converter 100, where the process of polygonrendering is further carried out at the pixel level (e.g., area fill andcolor interpolation for shading).

The microsequencer 97 operates in conjunction with various integer andfloating point arithmetic units, and may, for example, be an AMD 2910from Advanced Micro Devices.

I claim:
 1. In a graphic display system, a method of trimming apolygonal region in a two dimensional (J,K) space, the method comprisingthe steps of:a. selecting in a two dimensional (J,K) space an orderedcollection of original vertices describing an untrimmed polygonal regionin the (J,K) space; b. selecting two points P1 and P2 along a trimmingcurve in (J,K) space, P1 and P2 representing a directed line segmentP1P2 that is a portion of the trimming curve; c. setting a first flag toa first value when the directed line segment P1P2 is at least partiallywithin the untrimmed polygonal region and to a second value when thedirected line segment P1P2 is wholly outside and separated from theuntrimmed polygonal region; d. setting a second flag to a third valuewhen P1 is outside the untrimmed polygonal region and the directed linesegment P1P2 crosses into the untrimmed polygonal region, and to afourth value when P1 and the directed line segment P1P2 are otherwise;e. subsequent to setting the second flag to the third value in step (d),replacing the existing value of P1 with an adjusted value equal to thatof the point where the directed line segment P1P2 crosses into theuntrimmed polygonal region; f. setting a third flag to a fifth valuewhen the directed line segment P1P2 crosses out of the untrimmedpolygonal region and P2 is outside the untrimmed polygonal region, andto a sixth value when the directed line segment P1P2 and P2 areotherwise; g. subsequent to setting the third flag to the fifth value instep (f), replacing the existing value of P2 with an adjusted valueequal to that of the point where the directed line segment P1P2 crossesout of the untrimmed polygonal region; h. collecting, in the order inwhich they occur, an initial adjusted value for P1, any subsequentintervening value of P2 for which the first value for the first flag isassociated, followed by a final adjusted value for P2, into an orderedlist of points in (J,K) space along the trimming curve and which belongto the untrimmed polygonal region; i. combining the ordered collectionof original vertices selected in step (a) with the ordered listcollected in step (h) into a linked data structure of points in (J,K)space, the links in the data structure allowing the points to be visitedin a sequence corresponding to the order in which adjacent vertices of atrimmed polygonal region in (J,K) space are encountered during atraversal of the trimming curve and any remaining untrimmed boundariesbetween the original untrimmed vertices; j. traversing the linked datastructure to produce a sequence of adjacent vertices for a trimmedpolygonal region in the two dimensional (J,K) space; and k. displaying avisual image upon the graphics display system in accordance with thetrimmed polygon region.
 2. In a graphics display system, a method oftrimming a polygonal region in (u,v) parameter space, the vertices ofthe trimmed polygonal region used as arguments in mapping functions toproduce the vertices of a trimmed polygon in (X,Y,Z) space, the methodcomprising the steps of:a. selecting in a (u,v) parameter space anordered collection of original vertices describing an untrimmedpolygonal region in the (u,v) parameter space; b. selecting two pointsP1 and P2 along a trimming curve in (u,v) space, P1 and P2 representinga directed line segment P1P2 that is a portion of the trimming curve; c.setting a first flag to a first value when the directed line segmentP1P2 is at least partially within the untrimmed polygonal region and toa second value when the directed line segment P1P2 is wholly outside andseparated from the untrimmed polygonal region; d. setting a second flagto a third value when P1 is outside the untrimmed polygonal region andthe directed line segment P1P2 crosses into the untrimmed polygonalregion, and to a fourth value when P1 and the directed line segment P1P2are otherwise; e. subsequent to setting the second flag to the thirdvalue in step (d), replacing the existing value of P1 with an adjustedvalue equal to that of the point where the directed line segment P1P2crosses into the untrimmed polygonal region; f. setting a third flag toa fifth value when the directed line segment P1P2 crosses out of theuntrimmed polygonal region and P2 is outside the untrimmed polygonalregion, and to a sixth value when the directed line segment P1P2 and P2are otherwise; g. subsequent to setting the third flag to the fifthvalue in step (f), replacing the existing value of P2 with an adjustedvalue equal to that of the point where the directed line segment P1P2crosses out of the untrimmed polygonal region; h. collecting, in theorder in which they occur, an initial adjusted value for P1, anysubsequent intervening value of P2 for which the first value for thefirst flag is associated, followed by a final adjusted value for P2,into an ordered list of points in (u,v) parameter space along thetrimming curve and which belong to the untrimmed polygonal region; i.combining the ordered collection of original vertices selected in step(a) with the ordered list collected in step (h) into a linked datastructure of points in (u,v) parameter space, the links in the datastructure allowing the points to be visited in a sequence correspondingto the order in which adjacent vertices of a trimmed polygonal region in(u,v) space are encountered during a traversal of the trimming curve andany remaining untrimmed boundaries between the original untrimmedvertices; j. traversing the linked data structure to produce a sequenceof adjacent vertices for a trimmed polygonal region in the (u,v)parameter space; k. using the sequence of adjacent vertices in the (u,v)parameter space produced by the traversal of step (j) as arguments formapping functions to produce a corresponding sequence of coordinates in(X,Y,Z) space for a trimmed polygon; and l. displaying upon a graphicsdisplay system a visible image of the coordinates in (X,Y,Z) space of atrimmed polygonal region.
 3. A method of trimming a polygonal region in(u,v) parameter space, the vertices of the trimmed polygonal region usedas arguments in mapping functions to produce the vertices of a trimmedpolygon in (X,Y,Z) space and representing the surface of an object whosevisual projection onto an (X,Y) plane is electronically produced from amemory of at least xy-many addressable locations, each locationcontaining at least one pixel intensity value and a Z value, the methodcomprising the steps of:a. selecting in a (u,v) parameter space anordered collection of original vertices describing an untrimmedpolygonal region in the (u,v) parameter space; b. selecting two pointsP1 and P2 along a trimming curve in (u,v) space, P1 and P2 representinga directed line segment P1P2 that is a portion of the trimming curve; c.setting a first flag to a first value when the directed line segmentP1P2 is at least partially within the untrimmed polygonal region and toa second value when the directed line segment P1P2 is wholly outside andseparated from the untrimmed polygonal region; d. setting a second flagto a third value when P1 is outside the untrimmed polygonal region andthe directed line segment P1P2 crosses into the untrimmed polygonalregion, and to a fourth value when P1 and the directed line segment P1P2are otherwise; e. subsequent to setting the second flag to the thirdvalue in step (d), replacing the existing value of P1 with an adjustedvalue equal to that of the point where the directed line segment P1P2crosses into the untrimmed polygonal region; f. setting a third flag toa fifth value when the directed line segment P1P2 crosses out of theuntrimmed polygonal region and P2 is outside the untrimmed polygonalregion, and to a sixth value when the directed line segment P1P2 and P2are otherwise; g. subsequent to setting the third flag to the fifthvalue in step (f), replacing the existing value of P2 with an adjustedvalue equal to that of the point where the directed line segment P1P2crosses out of the untrimmed polygonal region; h. collecting, in theorder in which they occur, an initial adjusted value for P1, anysubsequent intervening value of P2 for which the first value for thefirst flag is associated, followed by a final adjusted value for P2,into an ordered list of points in (u,v) parameter space along thetrimming curve and which belong to the untrimmed polygonal region; i.combining the ordered collection of original vertices selected in step(a) with the ordered list collected in step (h) into a linked datastructure of points in (u,v) parameter space, the links in the datastructure allowing the points to be visited in a sequence correspondingto the order in which adjacent vertices of a trimmed polygonal region in(u,v) space are encountered during a traversal of the trimming curve andany remaining untrimmed boundaries between the original untrimmedvertices; j. traversing the linked data structure to produce a sequenceof adjacent vertices for a trimmed polygonal region in the (u,v)parameter space; k. using the sequence of adjacent vertices in the (u,v)parameter space produced by the traversal of step (j) as arguments formapping functions to produce a corresponding sequence of coordinates in(X,Y,Z) space for a trimmed polygon; l. addressing a memory of at leastxy-many addressable location according to the X and Y coordinate valuesproduced in step (k) and then storing in the addressed location at leastone pixel intensity value associated with the (X,Y,Z) coordinateproduced in step (k) and also storing in the addressed location the Zvalue produced in step (k); and m. electronically generating a visibledisplay in an XY plane according to the Z values and pixel intensityvalues stored by step (l).